On invariant sets and closed-loop boundedness of Lure-type nonlinear systems by LPV-embedding

We address the problem of achieving trajectory boundedness and computing ultimate bounds and invariant sets for Lure-type nonlinear systems with a sector-bounded nonlinearity. Our first contribution is to compare two systematic methods to compute invariant sets for Lure systems. In the first method, a linear-like bound is considered for the nonlinearity, and this bound is used to compute an invariant set by regarding the nonlinear system as a linear system with a nonlinear perturbation. In the second method, the sector-bounded nonlinearity is treated as a time-varying parameterised linear function with bounded parameter variations, and then invariant sets are computed by embedding the nonlinear system into a convex polytopic linear parameter varying (LPV) system. We show that under some conditions on the system matrices, these approaches give identical invariant sets, the LPV-embedding method being less conservative in the general case. The second contribution of the paper is to characterise a class of Lure systems, for which an appropriately designed linear state feedback achieves bounded trajectories of the closed-loop nonlinear system and allows for the computation of an invariant set via a simple, closed-form expression. The third contribution is to show that, for disturbances that are ‘aligned’ with the control input, arbitrarily small ultimate bounds on the system states can be achieved by assigning the eigenvalues of the linear part of the system with ‘large enough’ negative real part. We illustrate the results via examples of a pendulum system, a Josephson junction circuit and the well-known Chua circuit.